Appendix B — Vector/Matrix multiplication

R can be used to complete vector and matrix multiplication. The operator used for this is %*%. For example, for the following matrix and vector,

\[ \boldsymbol{X}=\begin{bmatrix}2&4&-1\\3&2&2\\1&2&-1\end{bmatrix},\,\,\,\,\boldsymbol{y}=\begin{bmatrix}1\\1\\3\end{bmatrix} \]

The solution to \(\boldsymbol{X}\times\boldsymbol{y}\) can be found using the following code.

R Code

X <- matrix(data = c(2, 4, -1, 3, 2, 2, 1, 2, -1), nrow = 3, byrow = TRUE)
y <- matrix(data = c(1, 1, 3), nrow = 3, byrow = TRUE)

X%*%y

Output

     [,1]
[1,]    3
[2,]   11
[3,]    0

If instead we wanted to calculate \(\boldsymbol{y}^\intercal\times\boldsymbol{X}\) we would first have to transpose y. Matrices can be transposed using the function t().

R Code

t(y)%*%X

Output

     [,1] [,2] [,3]
[1,]    8   12   -2

R can also be used to solve a system of equations. For example, we have just seen that,

\[ \begin{aligned} (2\times 1)+(4\times 1)+(-1\times 3)&=3\\ (3\times 1)+(2\times 1)+(2\times 3)&=11\\ (1\times 1)+(2\times 1)+(-1\times 3)&=0 \end{aligned} \]

But suppose we didn’t know the vector \(\boldsymbol{y}\), and instead were given the system of equations,

\[ \begin{aligned} 2x+4y-z&=3\\ 3x+2y+2z&=11\\ x+2y-z&=0 \end{aligned} \]

This can also be represented as,

\[ \boldsymbol{Xy}=\boldsymbol{z},\,\mbox{ where } \boldsymbol{X}=\begin{bmatrix}2&4&-1\\3&2&2\\1&2&-1\end{bmatrix},\,\,\,\,\boldsymbol{y}=\begin{bmatrix}x\\y\\z\end{bmatrix},\mbox{ and }\boldsymbol{z}=\begin{bmatrix}3\\11\\0\end{bmatrix} \]

We can then use R to solve this system of equations using the solve() function. We need to give solve() the matrix of coefficients, \(\boldsymbol{X}\), and the vector \(\boldsymbol{z}\).

R Code

z <- matrix(data = c(3, 11, 0), nrow = 3, byrow = TRUE)

solve(X, z)

Output

     [,1]
[1,]    1
[2,]    1
[3,]    3

This then gives the solution \(\boldsymbol{y}=\begin{bmatrix}1&1&3\end{bmatrix}^\intercal\), which is what we expected to see.

It is also possible to use the solve() function to simply find the inverse of a matrix. This is done by providing it with only one argument - the matrix to be inverted. For example, the code below returns the inverse of the matrix \(X\).

R Code

solve(X)

Output

      [,1]  [,2]  [,3]
[1,] -1.50  0.50  2.50
[2,]  1.25 -0.25 -1.75
[3,]  1.00  0.00 -2.00

We can see that,

\[ \boldsymbol{X}^{-1}=\begin{bmatrix}-1.50&0.50&2.50\\1.25&-0.25&-1.75\\1.00&0.00&-2.00\end{bmatrix} \]

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There are additional details and examples of using matrices in R in Sections 1.9.1 Arrays and Matrices and 1.9.2 Vector and Matrix Operations of Probability and Statistics with R.